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We obtain a new formula to relate the value of a Schur polynomial with variables $(x_1,\ldots,x_N)$ with values of Schur polynomials at $(1,\ldots,1)$. This allows to study the limit shape of perfect matchings on a square hexagon lattice with periodic weights and piecewise boundary conditions. In particular, when the edge weights satisfy certain conditions, asymptotics of the Schur function imply that the liquid region of the model in the scaling limit has multiple connected components, while the frozen boundary consists of disjoint cloud curves.

We show how to achieve perfect quantum state transfer and construct effective two-qubit gates between distant sites in engineered bosonic and fermionic networks. The Hamiltonian for the system can be determined by choosing an eigenvalue spectrum satisfying a certain condition, which is shown to be both sufficient and necessary in mirror-symmetrical networks. The natures of the effective two-qubit gates depend on the exchange symmetry for fermions and bosons. For fermionic networks, the gates are entangling (and thus universal for quantum computation). For bosonic networks, though the gates are not entangling, they allow two-way simultaneous communications. Protocols of entanglement generation in both bosonic and fermionic engineered networks are discussed.

We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of perfect Lie groups with a polynomial error term. Even for semisimple quotients, our result provides the first infinite family of examples where effective equidistribution with polynomial error rate is obtained for non-horospherical unipotent subgroups. The proof is based on the spectral gap of the ambient space, an effective closing lemma, Bourgain's discretized projection theorem, and a sub-modularity inequality in irreducible representation. The sub-modularity inequality is crucial to our proof and is of independent interest. As applications, we obtain effective estimates on distribution of lattice orbits on homogeneous spaces, as well as an effective version of the Oppenheim conjecture for indefinite quadratic forms with a polynomial error rate in all dimension $d \geq 3$.

This paper presents Pixel-Perfect Depth, a monocular depth estimation model based on pixel-space diffusion generation that produces high-quality, flying-pixel-free point clouds from estimated depth maps. Current generative depth estimation models fine-tune Stable Diffusion and achieve impressive performance. However, they require a VAE to compress depth maps into latent space, which inevitably introduces \textit{flying pixels} at edges and details. Our model addresses this challenge by directly performing diffusion generation in the pixel space, avoiding VAE-induced artifacts. To overcome the high complexity associated with pixel-space generation, we introduce two novel designs: 1) Semantics-Prompted Diffusion Transformers (SP-DiT), which incorporate semantic representations from vision foundation models into DiT to prompt the diffusion process, thereby preserving global semantic consistency while enhancing fine-grained visual details; and 2) Cascade DiT Design that progressively increases the number of tokens to further enhance efficiency and accuracy. Our model achieves the best performance among all published generative models across five benchmarks, and significantly outperform

We examine several ways to manipulate the loss in electromagnetic cloaks, based on transformation electromagnetics. It is found that, by utilizing inherent electric and magnetic losses of metamaterials, perfect wave absorption can be achieved based on several popular designs of electromagnetic cloaks. A practical implementation of the absorber, consisting of ten discrete layers of metamaterials, is proposed. The new devices demonstrate super-absorptivity over a moderate wideband range, suitable for both microwave and optical applications. It is corroborated that the device is functional with a subwavelength thickness and, hence, advantageous compared to the conventional absorbers.

In this work we contrast the behaviour of two spherically symmetric matter models in a class of spherically symmetric spacetimes which feature a weak null singularity. This class in particular contains spherically symmetric perturbations of subextremal Reissner-Nordström under the Einstein--Maxwell--scalar field system, a system for which a $C^2$ formulation of the strong cosmic censorship conjecture was proved by Luk-Oh, arXiv:1702.05715 and Dafermos, arXiv:1201.1797. Firstly, we consider the Cauchy problem of spherically symmetric dust falling into the weak null singularity (WNS) where the initial dust velocity is normal to a smooth spacelike curve with certain properties. We prove that the flow of the dust velocity does not experience any shell-crossing before or at the singularity, the velocity vector remains timelike, and that the dust energy density remains bounded as matter approaches the singularity. Secondly, we consider the characteristic initial value problem for stiff perfect fluid falling into the WNS. By relating the stiff fluid velocity and energy density to a scalar field satisfying the homogeneous linear wave equation, we prove that this energy density becomes infi

We propose a generalized CUR (GCUR) decomposition for matrix pairs $(A, B)$. Given matrices $A$ and $B$ with the same number of columns, such a decomposition provides low-rank approximations of both matrices simultaneously, in terms of some of their rows and columns. We obtain the indices for selecting the subset of rows and columns of the original matrices using the discrete empirical interpolation method (DEIM) on the generalized singular vectors. When $B$ is square and nonsingular, there are close connections between the GCUR of $(A, B)$ and the DEIM-induced CUR of $AB^{-1}$. When $B$ is the identity, the GCUR decomposition of $A$ coincides with the DEIM-induced CUR decomposition of $A$. We also show a similar connection between the GCUR of $(A, B)$ and the CUR of $AB^+$ for a nonsquare but full-rank matrix $B$, where $B^+$ denotes the Moore--Penrose pseudoinverse of $B$. While a CUR decomposition acts on one data set, a GCUR factorization jointly decomposes two data sets. The algorithm may be suitable for applications where one is interested in extracting the most discriminative features from one data set relative to another data set. In numerical experiments, we demonstrate th

We present block variants of the discrete empirical interpolation method (DEIM); as a particular application, we will consider a CUR factorization. The block DEIM algorithms are based on the concept of the maximum volume of submatrices and a rank-revealing QR factorization. We also present a version of the block DEIM procedures, which allows for adaptive choice of block size. The results of the experiments indicate that the block DEIM algorithms exhibit comparable accuracy for low-rank matrix approximation compared to the standard DEIM procedure. However, the block DEIM algorithms also demonstrate potential computational advantages, showcasing increased efficiency in terms of computational time.

A CUR factorization is often utilized as a substitute for the singular value decomposition (SVD), especially when a concrete interpretation of the singular vectors is challenging. Moreover, if the original data matrix possesses properties like nonnegativity and sparsity, a CUR decomposition can better preserve them compared to the SVD. An essential aspect of this approach is the methodology used for selecting a subset of columns and rows from the original matrix. This study investigates the effectiveness of \emph{one-round sampling} and iterative subselection techniques and introduces new iterative subselection strategies based on iterative SVDs. One provably appropriate technique for index selection in constructing a CUR factorization is the discrete empirical interpolation method (DEIM). Our contribution aims to improve the approximation quality of the DEIM scheme by iteratively invoking it in several rounds, in the sense that we select subsequent columns and rows based on the previously selected ones. Thus, we modify $A$ after each iteration by removing the information that has been captured by the previously selected columns and rows. We also discuss how iterative procedures fo

The discrete empirical interpolation method (DEIM) may be used as an index selection strategy for formulating a CUR factorization. A notable drawback of the original DEIM algorithm is that the number of column or row indices that can be selected is limited to the number of input singular vectors. We propose a new variant of DEIM, which we call L-DEIM, a combination of the strength of deterministic leverage scores and DEIM. This method allows for the selection of a number of indices greater than the number of available singular vectors. Since DEIM requires singular vectors as input matrices, L-DEIM is particularly attractive for example in big data problems when computing a rank-$k$-SVD approximation is expensive even for moderately small $k$ since it uses a lower-rank SVD approximation instead of the full rank-$k$ SVD. We empirically demonstrate the performance of L-DEIM, which despite its efficiency, may achieve comparable results to the original DEIM and even better approximations than some state-of-the-art methods.